HOTS : Straight Line

Prob 2.     If orthocentre of the triangle formed by ax2 + 2hxy + by2 = 0 and px + qy = 1 is (r, s) then prove that .

Sol.

Let ax2+2hxy + by2 = 0 be OA and OB.

Let OD and BE be perpendiculars on AB and OA respectively.

∴ Equation of OD is
…(1)

∴ H ≡ (pr, qr1) also lies on BE.

Now ,

∴ Equation of BE is

But (pr1, qr1) lies on BE

∴ r1(p + qm2) = (1 +  m1m2)/(p + m1q)

but m1 + m2 = – 2h/b  and m1m2 = a/b

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